Hadamard invertibility of linearly recursive sequences in several variables Original Research Article

A linearly recursive sequence in n variables is a tableau of scalars (ƒi1…in) for i1,i2,…, in ⩾ 0, such that for each 1 ⩽ i ⩽ n, all rows parallel to the ith axis satisfy a fixed linearly recursive relation hi(x) with constant coefficients. We show that such a tableau is Hadamard invertible (i.e., the tableau (1/ ƒi1… in) is linearly recursive) if and only if all ƒi1… in≠ 0, and each row is eventually an interlacing of geometric sequences. The procedure is effective, i.e., given a linearly recursive sequence ƒ = (ƒi1… in), it can be tested for Hadamard invertibility by a finite algorithm. These results extend the case n = 1 of Larson and Taft.